Optimization.  Theoretically analyzing the training process of most machine learning models is a challenging problem as the loss landscapes are highly non-convex. One approach to studying non-convex optimization problems is to use the Polyak-Łojasiewicz (PL) condition [1]}, which characterizes the local geometry of loss landscapes and ensures the existence of global minima. It is shown in [2]} that GD admits linear convergence for a class of optimization objective functions under the PL condition. In this paper,
we modify the original LGI to get length estimates of the optimization paths, which will be used to derive length-based generalization bounds.
